On minor-closed classes of matroids with exponential growth rate

Abstract

Let $\cM$ be a minor-closed class of matroids that does not contain arbitrarily long lines. The growth rate function, $h:\bN\rightarrow \bN$ of $\cM$ is given by $$h(n) = \max(|M|\, : \, M\in \cM, simple, rank-$n$).$$ The Growth Rate Theorem shows that there is an integer $c$ such that either: $h(n)\le cn$, or $\left(\genfrac{}{}{0ex}{}{n+1}{2}\right)\le h(n)\le c{n}^2$, or there is a prime-power $q$ such that $\frac{q^n-1}{q-1}\le h(n)\le c{q}^n$; this separates classes into those of linear density, quadratic density, and base-$q$ exponential density. For classes of base-$q$ exponential density that contain no $(q^2+1)$-point line, we prove that $h(n)=\frac{q^n-1}{q-1}$ for all sufficiently large $n$. We also prove that, for classes of base-$q$ exponential density that contain no $(q^2+q+1)$-point line, there exists $k\in\bN$ such that $h(n)=\frac{q^{n+k}-1}{q-1}-q\frac{q^{2k}-1}{q^2-1}$ for all sufficiently large $n$.